Proving Isomorphic Groups U(5) and U(10)

  • Thread starter gtfitzpatrick
  • Start date
  • Tags
    Groups
In summary, the conversation is about proving that the groups U(5) and U(10) are isomorphic. This can be done by showing that they have the same order and finding a cyclic element in both groups. A Cayley table is not necessary for this proof.
  • #1
gtfitzpatrick
379
0

Homework Statement



For any positive integern, let U(n) be the group of all positive integers less than n and relatively prime to n, under multiplication modulo n. Show the the Groups U(5) and u(10) are isomorphic

Homework Equations





The Attempt at a Solution



any 2 cyclic groups of the same size have to be isomorphic.
For the answer to this problem should i do out a caley table for both groups to show they are cyclic? is this enough along with my statement? i guess what I am saying is...does the question ask me to prove that 2 cyclic groups of the same size are isomorphic?
 
Physics news on Phys.org
  • #2


No, a Cayley table would be overkill! (although it would work).

I suggest you first work out what U(5) and U(10) is explicitely. Try to prove that they have the same order.
Then try to find a cyclic element in both groups.
 

1. What is the definition of isomorphic groups?

Isomorphic groups are groups that have the same structure and behave in the same way, despite having different elements. This means that they have the same number of elements, the same group operation, and the same identity element.

2. Can you prove that U(5) and U(10) are isomorphic?

Yes, we can prove that U(5) and U(10) are isomorphic by showing that they have the same structure and behave in the same way. This can be done by demonstrating a one-to-one correspondence between the elements of the two groups, and showing that the group operation and identity element are the same for both groups.

3. How do you find the isomorphism between U(5) and U(10)?

The isomorphism between U(5) and U(10) can be found by mapping the elements of one group to the corresponding elements of the other group. This can be done by using a mapping function that preserves the group operation and identity element.

4. What is the significance of proving isomorphic groups?

Proving isomorphic groups is important because it allows us to understand the relationship between different groups and how they behave. It also helps us to identify patterns and similarities between groups, which can be useful in solving problems and making connections in mathematics.

5. Can U(5) and U(10) be isomorphic to other groups?

No, U(5) and U(10) can only be isomorphic to each other because they are both groups of units modulo a certain number. Isomorphic groups must have the same underlying structure, so two groups of units modulo different numbers cannot be isomorphic.

Similar threads

Showing that cyclic groups of the same order are isomorphic
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Showing that GL(F_p) is isomorphic to an automorphism group
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
POTW Product of Two Finite Cyclic Groups
    • Math POTW for University Students
Replies
1
Views
536
Proving that an Abelian group of order pq is isomorphic to Z_pq
    • Calculus and Beyond Homework Help
Replies
5
Views
2K
If G is cyclic, and G is isomorphic to G', then G' is cycli
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
Characterizing subgroups of a cyclic group
    • Calculus and Beyond Homework Help
Replies
9
Views
1K
Is the group of positive rational numbers under * cyclic?
    • Calculus and Beyond Homework Help
Replies
3
Views
3K
Testing whether a binary structure is a group
    • Calculus and Beyond Homework Help
Replies
1
Views
877
Proving Closure and Identity for U(n)
    • Calculus and Beyond Homework Help
Replies
16
Views
4K
Proving that Aut(K), where K is cyclic, is abelian
    • Calculus and Beyond Homework Help
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top